1. Field of the Invention
The present invention relates to the general field of digital signal processing.
More particularly, the present invention relates to multiresolution analysis for signals or data of any dimensionality using a non-separable, radial frame, multi-resolution analysis in multidimensions.
2. Description of the Related Art
Digital signal processing, and, in general, the manipulation of information of all types by digitization, is at the heart of the computer-based approach to a vast range of problems in science, engineering, technology, modern economics modeling, data storage and retrieval, etc. There exist many robust approaches for problems which are intrinsically one-dimensional, and the theory of how one systematically parses the information content into small, manageable “chunks” is well developed. The essential idea is that information can be characterized either in a “physical” or “measurement” domain which, for example, we take to be the “time domain”, or in a complementary, mathematical domain referred to as the “Fourier domain” (which we shall refer to as the frequency domain). It is most useful if there are natural limitations on how much information in the measurement or time domain is required to characterize a given amount of information in the frequency domain.
In one-dimension (1-D), the best possible situation is when, e.g., only a finite range or “band” of data in the frequency domain is needed to characterize completely the underlying mathematical behavior. Such a situation is said to be “band-limited” and if one can capture, without any loss, all the frequency components contained in the band, the signal or phenomenon is exactly captured. Further, if the signal becomes “contaminated” by extraneous signals with frequencies outside the band range, then these can be eliminated by sending the signal through a “filter” that strains out everything except frequencies in the physical band range. This is accomplished mathematically simply by multiplying the signal (plus noise) by a function that is 1 for frequencies within the band-limit and 0 for all other frequencies. Such a filter is fundamental in all areas of signal processing and information analysis, and it is called “ideal filter” of “ideal window”.
The amazing fact is that when one must treat signals that are of higher dimension including the simplest case of (two dimensional (2-D) data), the only rigorous way that exists to create such an ideal window (and to shift or translate them to different frequency bands can be captured leading to a multiresolution) is by multiplying 1 -D ideal windows for each degree of freedom. Such products are said to be “separable” and they are inefficient for studying data sets or signals for which there is not a natural separation of information content along orthogonal directions. It is expected that in the absence of knowledge of such a directional bias in the signal, the best approach would treat the data in the most isotropic manner possible.
Thus, there is a need in the art of signal processing for the construction of improved multi-resolution analysis techniques for extracting information from complex scientific signals, especially processing techniques that involve non-separable, radial frame, multi-resolution analysis in one or more dimensions.